Let `S` be the set which contains all possible values of `l ,m ,n ,p ,q ,r` for which `A=[[l^2-3,p,0],[0,m^2-8,q],[r,0,n^2-15]]` be non-singular idempotent matrix. Then the sum of all the elements of the set `S` is ________.

Answer

0

Explanation

For idempotent matrix, `A^(2)=A`
`implies A^(-1) A^(2)=A^(-1)A" "( :' " A is nonsingular")`
or `A=I`
Thus, nonsingular idempotent matrix is always a unit matrix.
`:. L^(2)-3=1" "implies" "l= pm 2`
`m^(2)-8=1" "implies" "m= pm 3`
`n^(2)-15=1" "implies" "n= pm 4`
and `p=q=r=0`
Hence, the required sum is 0.